Video Transcript
Simplify the quotient of nine times two times the square root of six times four over
eight times the square root of two times 18 times the square root of 12 cubed.
We first recall that multiplication of real numbers is associative and commutative,
so we can multiply the factors in the numerator or denominator in any order. This allows us to write the numerator as nine times two times four multiplied by the
square root of six and the denominator as eight times 18 multiplied by the square
root of two times the square root of 12. The product of the rational numbers in the numerator is 72. And the product of the rational numbers in the denominator is 144.
We recall that for nonnegative real numbers 𝑎 and 𝑏, the product of the square root
of 𝑎 and the square root of 𝑏 equals the square root of the product of 𝑎 and
𝑏. Thus, the square root of two times the square root of 12 is equal to the square root
of 24. So, after simplifying the products in the numerator and denominator, we have 72 times
the square root of six over 144 times the square root of 24 cubed.
Next, we simplify the quotient of 72 over 144, which equals one over two. We recognize that the square root of six is a factor of the square root of 24. So we write the square root of 24 as the product of the square root of six and the
square root of four. This allows us to cancel the shared factor of the square root of six. We also know that the square root of four is equal to two. In the end, our expression simplifies to one-fourth cubed.
Finally, we recall the power of the quotient rule for exponents, which states that
the quotient of 𝑎 and 𝑏 to the power of 𝑛 is equal to the quotient of 𝑎 to the
power of 𝑛 over 𝑏 to the power of 𝑛, where 𝑎 is a real number, 𝑏 is a nonzero
real number, and 𝑛 is an integer. This means that one-fourth cubed is equal to one cubed over four cubed, which
simplifies to one over 64.